| Budget Amount *help |
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2003: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Research Abstract |
Recently much attentions have been paid to the field theory in the noncommutative space, where the coordinates are assumed to be noncommutative, [x,y]=-iθ. The corresponding field theory is obtained by replacing the ordinary product of two fields with the so-called star product. The simplest noncommutative space is expected to be realized in the 2-dimensional space.
Though it is studied extensively in particle theories, the only realisitic physical system governed by the noncommutative geometry is the quantum Hall (QH) system. The QH system is a world of planar electrons confined to the lowest Landau level under a strong magnetic field, where the x and y coordinates become noncommutative. As a result, when the electron possesses the SU(N) symmetry, the algebraic structure of the system becomes the SU(N) extension of the W_∞ algebra, which we have named the W_∞(N) algebra. We have constructed the quantum field theory of the QH system from this point of view.
Due to the noncommutativity we
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have shown that a quantum coherence develops spontaneously driven by the exchange interaction between neighboring electrons. This theoretical result is experimentally testable by observing topological solitons associated with the quantum coherence. It is intriguing that the topological soliton is a noncommutative soliton in QH systems.
Topological solitons have been known so far to be classical field configurations. As a main result, we have constructed a quantum mechanical state of a noncommutative soliton (skyrmion) by making a W_∞(N) rotation of a hole state. We have also derived an exact relation between the electron density and the topological density of a skyrmion. Furthermore, we calculated the excitation energy of a skyrmion both in the monolayer QH system and the bilayer QH system. We have also compared our results with the available experimental data. In this way we have analyzed physics in the noncommutaive space, taking an instance of the QH system, and demonstrated the validity of various concepts on the noncommutative geometry. Less
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